Chapter 27

Current and Resistance

Electric Current

Most practical applications of electricity deal with electric currents.

▪ The electric charges move through some region of space.

The resistor is a new element added to circuits.

Energy can be transferred to a device in an electric circuit.

The energy transfer mechanism is electrical transmission, TET.

Introduction

Electric Current

Electric current is the rate of flow of charge through some region of space.

The SI unit of current is the ampere (A).

▪ 1 A = 1 C / s

The symbol for electric current is I.

Section 27.1

Average Electric Current

Assume charges are moving

perpendicular to a surface of area A.

If ΔQ is the amount of charge that

passes through A in time Δt, then the

average current is

avg

Q

t

I

Section 27.1

Instantaneous Electric Current

If the rate at which the charge flows varies with time, the instantaneous current, I,

is defined as the differential limit of average current as Δt→0.

dQ

dtI

Section 27.1

Direction of Current

The charged particles passing through the surface could be positive, negative or

both.

It is conventional to assign to the current the same direction as the flow of

positive charges.

In an ordinary conductor, the direction of current flow is opposite the direction of

the flow of electrons.

It is common to refer to any moving charge as a charge carrier.

Section 27.1

Current and Drift Speed

Charged particles move through a cylindrical conductor of cross-sectional area A.

n is the number of mobile charge carriers per unit volume.

nAΔx is the total number of charge carriers in a segment.

Section 27.1

Current and Drift Speed, cont

The total charge is the number of carriers times the charge per carrier, q.

▪ ΔQ = (nAΔx)q

Assume the carriers move with a velocity parallel to the axis of the cylinder such that they experience a displacement in the x-direction.

If vd is the speed at which the carriers move, then

▪ vd = Δx / Δt and x = vd t

Rewritten: ΔQ = (nAvd Δt)q

Finally, current, Iave = ΔQ/Δt = nqvdA

vd is an average speed called the drift speed.

Section 27.1

Charge Carrier Motion in a Conductor

When a potential difference is applied across the conductor, an electric field is set up in the conductor which exerts an electric force on the electrons.

The motion of the electrons is no longer random.

The zigzag black lines represents the motion of a charge carrier in a conductor in the presence of an electric field.

▪ The net drift speed is small.

The sharp changes in direction are due to collisions.

The net motion of electrons is opposite the direction of the electric field.

Section 27.1

Motion of Charge Carriers, cont.

In the presence of an electric field, in spite of all the collisions, the charge carriers

slowly move along the conductor with a drift velocity,

The electric field exerts forces on the conduction electrons in the wire.

These forces cause the electrons to move in the wire and create a current.

dv

Section 27.1

Motion of Charge Carriers, final

The electrons are already in the wire.

They respond to the electric field set up by the battery.

The battery does not supply the electrons, it only establishes the electric field.

Section 27.1

Drift Velocity, Example

Assume a copper wire, with one free electron per atom contributed to the current.

The drift velocity for a 12-gauge copper wire carrying a current of 10.0 A is

2.23 x 10-4 m/s

▪ This is a typical order of magnitude for drift velocities.

Section 27.1

Current Density

J is the current density of a conductor.

It is defined as the current per unit area.

▪ J ≡ I / A = nqvd

▪ This expression is valid only if the current density is uniform and A is

perpendicular to the direction of the current.

J has SI units of A/m2

The current density is in the direction of the positive charge carriers.

Section 27.2

Conductivity

A current density and an electric field are established in a conductor whenever a potential difference is maintained across the conductor.

For some materials, the current density is directly proportional to the field.

The constant of proportionality, σ, is called the conductivity of the conductor.

Section 27.2

Ohm’s Law

Ohm’s law states that for many materials, the ratio of the current density to the electric field is a constant σ that is independent of the electric field producing the current.

▪ Most metals obey Ohm’s law

▪ Mathematically, J = σ E

▪ Materials that obey Ohm’s law are said to be ohmic

▪ Not all materials follow Ohm’s law

▪ Materials that do not obey Ohm’s law are said to be nonohmic.

Ohm’s law is not a fundamental law of nature.

Ohm’s law is an empirical relationship valid only for certain materials.

Section 27.2

Georg Simon Ohm

1789 -1854

German physicist

Formulated idea of resistance

Discovered the proportionalities now

known as forms of Ohm’s Law

Section 27.2

Resistance

In a conductor, the voltage applied across the ends of the conductor is

proportional to the current through the conductor.

The constant of proportionality is called the resistance of the conductor.

SI units of resistance are ohms (Ω).

▪ 1 Ω = 1 V / A

Resistance in a circuit arises due to collisions between the electrons carrying the

current with the fixed atoms inside the conductor.

VR

I

Section 27.2

Resistors

Most electric circuits use circuit

elements called resistors to control the

current in the various parts of the

circuit.

Stand-alone resistors are widely used.

▪ Resistors can be built into

integrated circuit chips.

Values of resistors are normally

indicated by colored bands.

▪ The first two bands give the first

two digits in the resistance value.

▪ The third band represents the

power of ten for the multiplier band.

▪ The last band is the tolerance.Section 27.2

Resistor Color Codes

Section 27.2

Resistor Color Code Example

Red (=2) and blue (=6) give the first two digits: 26

Green (=5) gives the power of ten in the multiplier: 105

The value of the resistor then is 26 x 105 Ω (or 2.6 MΩ)

The tolerance is 10% (silver = 10%) or 2.6 x 105 Ω

Section 27.2

Resistivity

The inverse of the conductivity is the resistivity:

▪ ρ = 1 / σ

Resistivity has SI units of ohm-meters (Ω . m)

Resistance is also related to resistivity:

R ρA

Section 27.2

Resistivity Values

Section 27.2

Resistance and Resistivity, Summary

Every ohmic material has a characteristic resistivity that depends on the

properties of the material and on temperature.

▪ Resistivity is a property of substances.

The resistance of a material depends on its geometry and its resistivity.

▪ Resistance is a property of an object.

An ideal conductor would have zero resistivity.

An ideal insulator would have infinite resistivity.

Section 27.2

Ohmic Material, Graph

An ohmic device

The resistance is constant over a wide range of voltages.

The relationship between current and voltage is linear.

The slope is related to the resistance.

Section 27.2

Nonohmic Material, Graph

Nonohmic materials are those whose resistance changes with voltage or current.

The current-voltage relationship is nonlinear.

A junction diode is a common example of a nonohmic device.

Section 27.2

Resistance of a Cable, Example

Assume the silicon between the

conductors to be concentric elements of

thickness dr.

The resistance of the hollow cylinder of

silicon is

2

ρdR dr

πrL

Section 27.2

Resistance of a Cable, Example, cont.

The total resistance across the entire thickness is

This is the radial resistance of the cable.

The calculated value is fairly high, which is desirable since you want the current to flow along the cable and not radially out of it.

2ln

b

a

ρ bR dR

πL a

Section 27.2

Electrical Conduction – A Model

Treat a conductor as a regular array of atoms plus a collection of free electrons.

▪ The free electrons are often called conduction electrons.

▪ These electrons become free when the atoms are bound in the solid.

In the absence of an electric field, the motion of the conduction electrons is random.

▪ Their speed is on the order of 106 m/s.

Section 27.3

Conduction Model, 2

When an electric field is applied, the conduction electrons are given a drift velocity.

Assumptions:

▪ The electron’s motion after a collision is independent of its motion before the collision.

▪ The excess energy acquired by the electrons in the electric field is transferred to the atoms of the conductor when the electrons and atoms collide.

▪ This causes the temperature of the conductor to increase.

Section 27.3

Conduction Model – Calculating the Drift Velocity

The force experienced by an electron is

From Newton’s Second Law, the acceleration is

Applying a motion equation

▪ Since the initial velocities are random, their average value is zero.

qF E

e

q

m m F E

a

f i f i

e

q= + t or = + t

m

Ev v a v v

Section 27.3

Conduction Model, 4

Let t be the average time interval between successive collisions.

The average value of the final velocity is the drift velocity.

This is also related to the current density: J = nqvd = (nq2E / me)t

▪ n is the number of charge carriers per unit volume.

,f avg d

e

q

mt

Ev v

Section 27.3

Conduction Model, final

Using Ohm’s Law, expressions for the conductivity and resistivity of a conductor can be found:

Note, according to this classical model, the conductivity and the resistivity do not depend on the strength of the field.

▪ This feature is characteristic of a conductor obeying Ohm’s Law.

2

2

1 e

e

mnq

m nq

t

t

Section 27.3

Resistance and Temperature

Over a limited temperature range, the resistivity of a conductor varies approximately linearly with the temperature.

▪ ρo is the resistivity at some reference temperature To

▪ To is usually taken to be 20° C

▪ α is the temperature coefficient of resistivity

▪ SI units of α are oC-1

The temperature coefficient of resistivity can be expressed as

[1 ( )]o oρ ρ α T T

1

o T

Section 27.4

Temperature Variation of Resistance

Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, you can find the effect of temperature on resistance.

R = Ro[1 + α(T - To)]

Use of this property enables precise temperature measurements through careful monitoring of the resistance of a probe made from a particular material.

Section 27.4

Resistivity and Temperature, Graphical View

For some metals, the resistivity is

nearly proportional to the temperature.

A nonlinear region always exists at very

low temperatures.

The resistivity usually reaches some

finite value as the temperature

approaches absolute zero.

Section 27.4

Residual Resistivity

The residual resistivity near absolute zero is caused primarily by the

collisions of electrons with impurities and imperfections in the metal.

High temperature resistivity is predominantly characterized by collisions

between the electrons and the metal atoms.

▪ This is the linear range on the graph.

Section 27.4

Semiconductors

Semiconductors are materials that exhibit a decrease in resistivity with an

increase in temperature.

α is negative

There is an increase in the density of charge carriers at higher temperatures.

Section 27.4

Superconductors

A class of materials and compounds whose resistances fall to virtually zero below a certain temperature, TC.

▪ TC is called the critical temperature.

The graph is the same as a normal metal above TC, but suddenly drops to zero at TC.

Section 27.5

Superconductors, cont

The value of TC is sensitive to:

▪ chemical composition

▪ pressure

▪ molecular structure

Once a current is set up in a superconductor, it persists without any applied voltage.

▪ Since R = 0

Section 27.5

Superconductor Application

An important application of superconductors is a superconducting magnet.

The magnitude of the magnetic field is about 10 times greater than a normal electromagnet.

These magnets are being considered as a means of storing energy.

Are currently used in MRI units

Section 27.5

Electrical Power

Assume a circuit as shown

The entire circuit is the system.

As a charge moves from a to b, the electric potential energy of the system increases by QV.

▪ The chemical energy in the battery must decrease by this same amount.

This electric potential energy is transformed into internal energy in the resistor.

▪ Corresponds to increased vibrational motion of the atoms in the resistor

Section 27.6

Electric Power, 2

The resistor is normally in contact with the air, so its increased temperature will

result in a transfer of energy by heat into the air.

The resistor also emits thermal radiation.

After some time interval, the resistor reaches a constant temperature.

▪ The input of energy from the battery is balanced by the output of energy by

heat and radiation.

The rate at which the system’s potential energy decreases as the charge passes

through the resistor is equal to the rate at which the system gains internal energy

in the resistor.

The power is the rate at which the energy is delivered to the resistor.

Section 27.6

Electric Power, final

The power is given by the equation P = I ΔV.

Applying Ohm’s Law, alternative expressions can be found:

Units: I is in A, R is in Ω, ΔV is in V, and P is in W

2

2 V

P V RR

I I

Section 27.6

Some Final Notes About Current

A single electron is moving at the drift velocity in the circuit.

▪ It may take hours for an electron to move completely around a circuit.

The current is the same everywhere in the circuit.

▪ Current is not “used up” anywhere in the circuit

The charges flow in the same rotational sense at all points in the circuit.

Electric Power Transmission

Real power lines have resistance.

Power companies transmit electricity at high voltages and low currents to

minimize power losses.